$$x^{\frac{1}{n}}=\sqrt[n]{x}$$
is usually true by definition as those are two notations for the same function.

Not so sure that you can just say they are two notations for the same function.

You can define $$\sqrt(x)$$ as that number when multiplied by itself is $$x$$, ie square root is inverse of squaring.

However when dealing with powers you cannot just say well $$x^{\frac{1}{2}}$$ is a new notation for square root, it is the extension of the notation of powers from positive integers to rationals and its interpretation has yet to be determined.
Using the usual rules for powers then

Not so sure that you can just say they are two notations for the same function.

You can define $$\sqrt(x)$$ as that number when multiplied by itself is $$x$$, ie square root is inverse of squaring.

However when dealing with powers you cannot just say well $$x^{\frac{1}{2}}$$ is a new notation for square root, it is the extension of the notation of powers from positive integers to rationals and its interpretation has yet to be determined.
Using the usual rules for powers then

I personally would think of the rules for exponent to be defined so that ##\{a^x|x\in\mathbb{R}\}## with ##a## being some constant that is greater than zero and not 1, is a group under multiplication. From this, like jing2178 shows, it is easy to prove that ##x^{1/n}## is a number such that, when raised to the ##n^{th}## power, one gets ##x##. From there it is clear that the notation ##\sqrt[n]{x}## and ##x^{1/n}## mean the same thing (ignoring discussions of principle roots).

It is equally valid to set the definition that those are the same notation and then show that the exponent rules work out to keep the group structure. To me the other way makes me feel happier inside

I personally would think of the rules for exponent to be defined so that ##\{a^x|x\in\mathbb{R}\}## with ##a## being some constant that is greater than zero and not 1, is a group under multiplication. From this, like jing2178 shows, it is easy to prove that ##x^{1/n}## is a number such that, when raised to the ##n^{th}## power, one gets ##x##. From there it is clear that the notation ##\sqrt[n]{x}## and ##x^{1/n}## mean the same thing (ignoring discussions of principle roots).